The weighted inequality for cross-intersecting pairs with profile-dependent skewness

Let $(A_1,B_1),\dots,(A_m,B_m)$ be pairs of finite non-empty subsets of $\mathbb{N}$. Write $a_i=|A_i|$ and $b_i=|B_i|$ for $1\leq i\leq m$. Assume that

$$A_i\cap B_i=\emptyset\quad (1\leq i\leq m),$$ $$A_i\cap B_j\ne\emptyset\quad (1\leq i<j\leq m),$$

and $A_i\cap B_j\ne\emptyset$ whenever $|A_i|\ne|A_j|$ or $|B_i|\ne|B_j|$. Weighted cross-intersection conjecture. Then

$$\sum_{i=1}^m\frac{1}{\binom{a_i+b_i}{a_i}}\leq 1.$$

This asks whether full cross-intersection for pairs with distinct profiles, together with only the stated ordered condition for pairs of the same profile, still implies the Bollobás-type weighted bound. The surrounding examples show that relaxing the ordering condition can allow weighted sums greater than $1$, so the validity of this particular relaxation remains unresolved.